Rotation given by a cosine-sine pair. More...

#include <src/eigen/Eigen/src/Jacobi/Jacobi.h>

Public Types
typedef NumTraits< Scalar >::Real	RealScalar

Public Member Functions
	JacobiRotation ()

	JacobiRotation (const Scalar &c, const Scalar &s)

Scalar &	c ()

Scalar	c () const

Scalar &	s ()

Scalar	s () const

JacobiRotation	operator* (const JacobiRotation &other)

JacobiRotation	transpose () const

JacobiRotation	adjoint () const

template<typename Derived >
bool	makeJacobi (const MatrixBase< Derived > &, Index p, Index q)

bool	makeJacobi (const RealScalar &x, const Scalar &y, const RealScalar &z)

void	makeGivens (const Scalar &p, const Scalar &q, Scalar *r=0)

Protected Member Functions
void	makeGivens (const Scalar &p, const Scalar &q, Scalar *r, internal::true_type)

void	makeGivens (const Scalar &p, const Scalar &q, Scalar *r, internal::false_type)

Protected Attributes
Scalar	m_c

Scalar	m_s

Detailed Description

template<typename Scalar>
class Eigen::JacobiRotation< Scalar >

Rotation given by a cosine-sine pair.

\jacobi_module

This class represents a Jacobi or Givens rotation. This is a 2D rotation in the plane J of angle $\theta$ defined by its cosine c and sine s as follow: $J = \left ( \begin{array}{cc} c & \overline s \\ -s & \overline c \end{array} \right )$

You can apply the respective counter-clockwise rotation to a column vector v by applying its adjoint on the left: $ v = J^* v $ that translates to the following Eigen code:

v.applyOnTheLeft(J.adjoint());

See also: MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()

Member Typedef Documentation

◆ RealScalar

template<typename Scalar >

typedef NumTraits<Scalar>::Real Eigen::JacobiRotation< Scalar >::RealScalar

Constructor & Destructor Documentation

◆ JacobiRotation() [1/2]

template<typename Scalar >

Eigen::JacobiRotation< Scalar >::JacobiRotation ( )

inline

Default constructor without any initialization.

40{}

Referenced by Eigen::JacobiRotation< Scalar >::adjoint(), Eigen::JacobiRotation< Scalar >::operator*(), and Eigen::JacobiRotation< Scalar >::transpose().

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◆ JacobiRotation() [2/2]

template<typename Scalar >

Eigen::JacobiRotation< Scalar >::JacobiRotation	(	const Scalar &	c,
		const Scalar &	s
	)

inline

Construct a planar rotation from a cosine-sine pair (c, s).

43: m_c(c), m_s(s) {}

Eigen::JacobiRotation::m_c

Scalar m_c

Definition Jacobi.h:74

Eigen::JacobiRotation::s

Scalar & s()

Definition Jacobi.h:47

Eigen::JacobiRotation::c

Scalar & c()

Definition Jacobi.h:45

Eigen::JacobiRotation::m_s

Scalar m_s

Definition Jacobi.h:74

Member Function Documentation

◆ adjoint()

template<typename Scalar >

JacobiRotation Eigen::JacobiRotation< Scalar >::adjoint ( ) const

inline

Returns the adjoint transformation

62{ using numext::conj; return JacobiRotation(conj(m_c), -m_s); }

Eigen::JacobiRotation::JacobiRotation

JacobiRotation()

Definition Jacobi.h:40

References Eigen::JacobiRotation< Scalar >::JacobiRotation(), Eigen::JacobiRotation< Scalar >::m_c, and Eigen::JacobiRotation< Scalar >::m_s.

Referenced by Eigen::RealQZ< _MatrixType >::hessenbergTriangular(), Eigen::RealQZ< _MatrixType >::pushDownZero(), Eigen::ComplexSchur< _MatrixType >::reduceToTriangularForm(), Eigen::internal::svd_precondition_2x2_block_to_be_real< MatrixType, QRPreconditioner, true >::run(), Eigen::RealQZ< _MatrixType >::splitOffTwoRows(), Eigen::RealSchur< _MatrixType >::splitOffTwoRows(), and Eigen::RealQZ< _MatrixType >::step().

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◆ c() [1/2]

template<typename Scalar >

Scalar & Eigen::JacobiRotation< Scalar >::c ( )

inline

45{ return m_c; }

References Eigen::JacobiRotation< Scalar >::m_c.

Referenced by Eigen::MatrixBase< Homogeneous< MatrixType, _Direction > >::applyOnTheRight(), Eigen::internal::real_2x2_jacobi_svd(), Eigen::internal::svd_precondition_2x2_block_to_be_real< MatrixType, QRPreconditioner, true >::run(), and Eigen::internal::tridiagonal_qr_step().

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◆ c() [2/2]

template<typename Scalar >

Scalar Eigen::JacobiRotation< Scalar >::c ( ) const

inline

46{ return m_c; }

References Eigen::JacobiRotation< Scalar >::m_c.

◆ makeGivens() [1/3]

template<typename Scalar >

void Eigen::JacobiRotation< Scalar >::makeGivens	(	const Scalar &	p,
		const Scalar &	q,
		Scalar *	r,
		internal::false_type
	)

protected

{
  using std::sqrt;
  using std::abs;
  if(q==Scalar(0))
  {
    m_c = p<Scalar(0) ? Scalar(-1) : Scalar(1);
    m_s = Scalar(0);
    if(r) *r = abs(p);
  }
  else if(p==Scalar(0))
  {
    m_c = Scalar(0);
    m_s = q<Scalar(0) ? Scalar(1) : Scalar(-1);
    if(r) *r = abs(q);
  }
  else if(abs(p) > abs(q))
  {
    Scalar t = q/p;
    Scalar u = sqrt(Scalar(1) + numext::abs2(t));
    if(p<Scalar(0))
      u = -u;
    m_c = Scalar(1)/u;
    m_s = -t * m_c;
    if(r) *r = p * u;
  }
  else
  {
    Scalar t = p/q;
    Scalar u = sqrt(Scalar(1) + numext::abs2(t));
    if(q<Scalar(0))
      u = -u;
    m_s = -Scalar(1)/u;
    m_c = -t * m_s;
    if(r) *r = q * u;
  }
 
}

References sqrt().

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◆ makeGivens() [2/3]

template<typename Scalar >

void Eigen::JacobiRotation< Scalar >::makeGivens	(	const Scalar &	p,
		const Scalar &	q,
		Scalar *	r,
		internal::true_type
	)

protected

{
  using std::sqrt;
  using std::abs;
  using numext::conj;
  
  if(q==Scalar(0))
  {
    m_c = numext::real(p)<0 ? Scalar(-1) : Scalar(1);
    m_s = 0;
    if(r) *r = m_c * p;
  }
  else if(p==Scalar(0))
  {
    m_c = 0;
    m_s = -q/abs(q);
    if(r) *r = abs(q);
  }
  else
  {
    RealScalar p1 = numext::norm1(p);
    RealScalar q1 = numext::norm1(q);
    if(p1>=q1)
    {
      Scalar ps = p / p1;
      RealScalar p2 = numext::abs2(ps);
      Scalar qs = q / p1;
      RealScalar q2 = numext::abs2(qs);
 
      RealScalar u = sqrt(RealScalar(1) + q2/p2);
      if(numext::real(p)<RealScalar(0))
        u = -u;
 
      m_c = Scalar(1)/u;
      m_s = -qs*conj(ps)*(m_c/p2);
      if(r) *r = p * u;
    }
    else
    {
      Scalar ps = p / q1;
      RealScalar p2 = numext::abs2(ps);
      Scalar qs = q / q1;
      RealScalar q2 = numext::abs2(qs);
 
      RealScalar u = q1 * sqrt(p2 + q2);
      if(numext::real(p)<RealScalar(0))
        u = -u;
 
      p1 = abs(p);
      ps = p/p1;
      m_c = p1/u;
      m_s = -conj(ps) * (q/u);
      if(r) *r = ps * u;
    }
  }
}

References sqrt().

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◆ makeGivens() [3/3]

template<typename Scalar >

void Eigen::JacobiRotation< Scalar >::makeGivens	(	const Scalar &	p,
		const Scalar &	q,
		Scalar *	r = `0`
	)

Makes *this as a Givens rotation G such that applying $ G^* $ to the left of the vector $V = \left ( \begin{array}{c} p \\ q \end{array} \right )$ yields: $G^* V = \left ( \begin{array}{c} r \\ 0 \end{array} \right )$ .

The value of r is returned if r is not null (the default is null). Also note that G is built such that the cosine is always real.

Example:

Output:

This function implements the continuous Givens rotation generation algorithm found in Anderson (2000), Discontinuous Plane Rotations and the Symmetric Eigenvalue Problem. LAPACK Working Note 150, University of Tennessee, UT-CS-00-454, December 4, 2000.

See also: MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()

{
  makeGivens(p, q, r, typename internal::conditional<NumTraits<Scalar>::IsComplex, internal::true_type, internal::false_type>::type());
}

Referenced by Eigen::RealQZ< _MatrixType >::hessenbergTriangular(), Eigen::internal::llt_rank_update_lower(), Eigen::RealQZ< _MatrixType >::pushDownZero(), Eigen::ComplexSchur< _MatrixType >::reduceToTriangularForm(), Eigen::RealQZ< _MatrixType >::splitOffTwoRows(), Eigen::RealSchur< _MatrixType >::splitOffTwoRows(), Eigen::RealQZ< _MatrixType >::step(), and Eigen::internal::tridiagonal_qr_step().

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◆ makeJacobi() [1/2]

template<typename Scalar >

template<typename Derived >

bool Eigen::JacobiRotation< Scalar >::makeJacobi	(	const MatrixBase< Derived > &	m,
		Index	p,
		Index	q
	)

inline

Makes *this as a Jacobi rotation J such that applying J on both the right and left sides of the 2x2 selfadjoint matrix $B = \left ( \begin{array}{cc} \text{this}_{pp} & \text{this}_{pq} \\ (\text{this}_{pq})^* & \text{this}_{qq} \end{array} \right )$ yields a diagonal matrix $ A = J^* B J $

Example:

Output:

See also: JacobiRotation::makeJacobi(RealScalar, Scalar, RealScalar), MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()

{
  return makeJacobi(numext::real(m.coeff(p,p)), m.coeff(p,q), numext::real(m.coeff(q,q)));
}

Referenced by Eigen::internal::real_2x2_jacobi_svd().

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◆ makeJacobi() [2/2]

template<typename Scalar >

bool Eigen::JacobiRotation< Scalar >::makeJacobi	(	const RealScalar &	x,
		const Scalar &	y,
		const RealScalar &	z
	)

Makes *this as a Jacobi rotation J such that applying J on both the right and left sides of the selfadjoint 2x2 matrix $B = \left ( \begin{array}{cc} x & y \\ \overline y & z \end{array} \right )$ yields a diagonal matrix $ A = J^* B J $

See also: MatrixBase::makeJacobi(const MatrixBase<Derived>&, Index, Index), MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()

{
  using std::sqrt;
  using std::abs;
  RealScalar deno = RealScalar(2)*abs(y);
  if(deno < (std::numeric_limits<RealScalar>::min)())
  {
    m_c = Scalar(1);
    m_s = Scalar(0);
    return false;
  }
  else
  {
    RealScalar tau = (x-z)/deno;
    RealScalar w = sqrt(numext::abs2(tau) + RealScalar(1));
    RealScalar t;
    if(tau>RealScalar(0))
    {
      t = RealScalar(1) / (tau + w);
    }
    else
    {
      t = RealScalar(1) / (tau - w);
    }
    RealScalar sign_t = t > RealScalar(0) ? RealScalar(1) : RealScalar(-1);
    RealScalar n = RealScalar(1) / sqrt(numext::abs2(t)+RealScalar(1));
    m_s = - sign_t * (numext::conj(y) / abs(y)) * abs(t) * n;
    m_c = n;
    return true;
  }
}

References sqrt().

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◆ operator*()

template<typename Scalar >

JacobiRotation Eigen::JacobiRotation< Scalar >::operator* ( const JacobiRotation< Scalar > & other )

inline

Concatenates two planar rotation

    {
      using numext::conj;
      return JacobiRotation(m_c * other.m_c - conj(m_s) * other.m_s,
                            conj(m_c * conj(other.m_s) + conj(m_s) * conj(other.m_c)));
    }

References Eigen::JacobiRotation< Scalar >::JacobiRotation(), Eigen::JacobiRotation< Scalar >::m_c, and Eigen::JacobiRotation< Scalar >::m_s.

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◆ s() [1/2]

template<typename Scalar >

Scalar & Eigen::JacobiRotation< Scalar >::s ( )

inline

47{ return m_s; }

References Eigen::JacobiRotation< Scalar >::m_s.

Referenced by Eigen::MatrixBase< Homogeneous< MatrixType, _Direction > >::applyOnTheRight(), Eigen::internal::real_2x2_jacobi_svd(), Eigen::internal::svd_precondition_2x2_block_to_be_real< MatrixType, QRPreconditioner, true >::run(), and Eigen::internal::tridiagonal_qr_step().

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◆ s() [2/2]

template<typename Scalar >

Scalar Eigen::JacobiRotation< Scalar >::s ( ) const

inline

48{ return m_s; }

References Eigen::JacobiRotation< Scalar >::m_s.

◆ transpose()

template<typename Scalar >

JacobiRotation Eigen::JacobiRotation< Scalar >::transpose ( ) const

inline

Returns the transposed transformation

59{ using numext::conj; return JacobiRotation(m_c, -conj(m_s)); }

References Eigen::JacobiRotation< Scalar >::JacobiRotation(), Eigen::JacobiRotation< Scalar >::m_c, and Eigen::JacobiRotation< Scalar >::m_s.

Referenced by Eigen::MatrixBase< Derived >::applyOnTheRight(), Eigen::RealQZ< _MatrixType >::compute(), Eigen::JacobiSVD< _MatrixType, QRPreconditioner >::compute(), and Eigen::internal::real_2x2_jacobi_svd().

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Member Data Documentation

◆ m_c

template<typename Scalar >

Scalar Eigen::JacobiRotation< Scalar >::m_c

protected

Referenced by Eigen::JacobiRotation< Scalar >::adjoint(), Eigen::JacobiRotation< Scalar >::c(), Eigen::JacobiRotation< Scalar >::c(), Eigen::JacobiRotation< Scalar >::operator*(), and Eigen::JacobiRotation< Scalar >::transpose().

◆ m_s

template<typename Scalar >

Scalar Eigen::JacobiRotation< Scalar >::m_s

protected

Referenced by Eigen::JacobiRotation< Scalar >::adjoint(), Eigen::JacobiRotation< Scalar >::operator*(), Eigen::JacobiRotation< Scalar >::s(), Eigen::JacobiRotation< Scalar >::s(), and Eigen::JacobiRotation< Scalar >::transpose().

The documentation for this class was generated from the following files:

src/eigen/Eigen/src/Core/util/ForwardDeclarations.h
src/eigen/Eigen/src/Jacobi/Jacobi.h

Public Types

Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

Member Typedef Documentation

◆ RealScalar

Constructor & Destructor Documentation

◆ JacobiRotation() [1/2]

◆ JacobiRotation() [2/2]

Member Function Documentation

◆ adjoint()

◆ c() [1/2]

◆ c() [2/2]

◆ makeGivens() [1/3]

◆ makeGivens() [2/3]

◆ makeGivens() [3/3]

◆ makeJacobi() [1/2]

◆ makeJacobi() [2/2]

◆ operator*()

◆ s() [1/2]

◆ s() [2/2]

◆ transpose()

Member Data Documentation

◆ m_c

◆ m_s